I already gave an example of the kinds of problems that were given to Jewish people at the oral entrance exam to the math department of Moscow State University. In fact, I have a whole page with a collection of such problems, called Jewish problems or Coffins. That page was one of the first pages I created when I started my website more than ten years ago.

When my son Alexey was in high school, I asked him to help me type these problems into a file and to recover their solutions from my more than laconic notes, and solve the problems that I didn’t have notes for. He did the job, but the file was lying dormant on my computer. Recently I resurrected the file and we prepared some of the solutions for a publication.

The problems that were given during these exams were very different in flavor: some were intentionally ambiguous questions, some were just plain hard, some had impossible premises. In our joint paper “Jewish Problems” we presented problems with a special flavor. These are problems that have a short and “simple” solution, that is nonetheless very difficult to find. This way the math department of MSU was better protected from appeals and complaints.

Try the following problem from our paper:

Find all real functions of real variable F(x) such that for any x and y the following inequality holds: F(x) − F(y) ≤ (x − y)2.

I will give a talk on the subject for UMA at MIT on October 18, at 5pm.

10 Comments

Andrei Zelevinsky:

Tanya: 1) I thought using derivatives was not allowed on entrance exams. 2) Problem 4: this is an equation, not an inequality. 3) Solution of Problem 18: I don’t think the part “We see that they are getting smaller very fast” would be accepted on the exam …

Count Iblis:

Joseph:

No need for derivatives on this one.

Since x and y are “switchable”, we are basically given that |F(x)-F(y)|<=(x-y)^2 for all x and y. Pick any two distinct real numbers x and y (assume x<y, without loss of generality), and assume that the distance between them is m.

Now choose any natural number n, and partition the interval [x,y] into n equal subintervals. Let x_0=x, x_1=x+(m/n), x_2=x+2(m/n), and so on, up to x_n=y. We then have: